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On o-minimal homotopy groups

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2010
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Oxford University Press
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We work over an o-minimal expansion of a real closed field. The o-minimal homotopy groups of a definable set are defined naturally using definable continuous maps. We prove that any two semialgebraic maps which are definably homotopic are also semialgebraically homotopic. This result together with known results on semialgebraic homotopy allows us to develop an o-minimal homotopy theory. In particular, we obtain o-minimal versions of the Hurewicz theorems and the Whitehead theorem.
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Álgebra
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1201 Álgebra
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[1] E. Baro, Normal triangulations in o-minimal structures, preprint, 15pp.,2007, www.uam.es/elias.baro/articulos.html. [2] E. Baro and M. Otero, Locally de nable homotopy, preprint, 33pp., 2008, www.uam.es/elias.baro/articulos.html. [3] A. Berarducci, M. Mamino and M. Otero, Higher homotopy of groups definable in o-minimal structures, 2008 Preprint, arXiv:0809.4940 [math.LO]. [4] A. Berarducci and M. Otero, o-minimal fundamental group, homology and manifolds, J. London Math. Soc. (2) 65 (2002), no. 2, 257-270. [5] A. Berarducci and M. Otero, Transfer methods for o-minimal topology, J. Symb. Log. 68 (2003) 785-794. [6] L. van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, 248, Cambridge University Press, 1998. [7] H. Delfs and M. Knebusch, Separation, retractions and homotopy extension in semialgebraic spaces, Paci_c J. Math. 114 (1984), no. 1, 47-71. [8] H. Delfs and M. Knebusch, Locally semialgebraic spaces, Lecture Notes in Mathematics, 1173, Springer-Verlag, Berlin, 1985. [9] M.Edmundo and M. Otero, Definably compact abelian groups, J. Math.Log. 4 (2004), no. 2, 163-180. [10] A. Hatcher, Algebraic topology, Cambridge University Press, 2002. [11] S. Hu, Homotopy theory, Pure and Applied Mathematics, Vol. VIII Academic Press, New York-London 1959. [12] A. Piekosz, O-minimal homotopy and generalized (co)homology, preprint, 2008. [13] A.Woerheide, O-minimal homology, PhD Thesis, University of Illinois at Urbana-Champaign, 1996.
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https://hdl.handle.net/20.500.14352/42048
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